Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory ${\mathcal {A}}$ of a category ${\mathcal {B}}$ is said to be isomorphism-closed or replete if every ${\mathcal {B}}$-isomorphism ${\displaystyle h:A\to B}$ with $A\in {\mathcal {A}}$ belongs to ${\displaystyle {\mathcal {A}}.}$ This implies that both $B$ and ${\displaystyle h^{-1}:B\to A}$ belong to ${\mathcal {A}}$ as well.

A subcategory which is isomorphism-closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every ${\mathcal {B}}$-object which is isomorphic to an ${\mathcal {A}}$-object is also an ${\mathcal {A}}$-object.

This condition is very natural. E.g. in the category of topological spaces one usually studies properties which are invariant under homeomorphisms – so called topological properties. Every topological property corresponds to a strictly full subcategory of ${\displaystyle \mathbf {Top} .}$

References

This article incorporates material from Isomorphism-closed subcategory on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.